International Journal of Computer Applications (0975 8887)

Volume 112 No. 1, February 2015

Artificial Bee Colony Algorithm for Solving OPF Problem

Considering the Valve Point EffectSouhil Mouassa

Tarek Bouktir

Department of Electrical Engineering

University of Chlef

Department of Electrical Engineering

University of Setif 1

ABSTRACTArtificial Bee Colony Algorithm (ABC) is a viableoptimization algorithm, based on simulating of the foragingbehavior of honey bee swarm. This paper is examined theability of Artificial Bee Colony algorithm for solving theOptimal Power Flow (OPF) problem considering the valvepoint effects in a power systems. The objective functionsconsidered are: fuel cost minimization, the valve point effectand multi-fuel of generation units. The proposed algorithm isapplied to determine the optimal settings of OPF problemcontrol variables. The feasibility of the proposed algorithmhas been tested on the IEEE 30-bus and IEEE-57 bus testsystems, with different objective functions. Several caseswere investigated to test and validate the robustness of theproposed algorithm in finding the optimal solution or the nearoptimal solution for each objective. Moreover, the obtainedresults are compared with those available recently in theliterature. Therefore, the ABC algorithm could be a usefulalgorithm for implementation in solving the OPF problem.

1. INTRODUCTIONThe optimal power flow is one of important optimizationproblem in the power system. It was introduced first time in1968 by Dommel and Tinney [1], and it is currentlyconsidered one of the most useful tools for modern powersystems operations and planning [2, 3, 4, 5], because it is abackbone of power system. In general, the OPF is a nonlinearprogramming (NLP) problem that determines the optimalcontrol set points of the system to minimize a given objectivefunction, subject at the same time to equality and inequalityconstraints imposed by the power system. In other words, is todetermine the optimal combination of real power generations,voltage magnitudes, shunt capacitor, and transformer tapsettings to minimize a desired objective function. Severalconventional optimization methods such as linearprogramming (LP), interior point method, reduced gradientmethod and Newton method (Huneault & Galiana, 1991;Momoh, Adapa, & El-Hawary, 1999[6]) have been applied tosolve OPF problem assuming convex, differentiable and linearcost function. But unfortunately, these methods face problemsin yielding optimal solution in practical systems due tononlinear and non-convex characteristic [7] like valve pointeffects loading in fossil fuel burning plants [8-3]. Hence, itbecomes essential to develop optimization algorithms that arecapable of overcoming these drawbacks and handling suchdifficulties. Complex constrained optimization problems havebeen solved by many population-based optimizationalgorithms in the recent years. These techniques have beensuccessfully applied to non-convex, non-smooth and nondifferentiable optimization problems. Some of the populationbased optimization methods are genetic algorithm [3], Particle

Swarm Optimization [9], Differential

Evolutionary Programming [11].

Evolution

[10]

Recently, a new evolutionary computation algorithm, based

on simulating the foraging behavior of honey bee swarmcalled Artificial Bee Colony (ABC), has been developedand introduced by Karaboga in 2005 for real-parameteroptimization. Since ABC algorithm is simple in concept, easyto implement, and has fewer control parameters, it has beenwidely used in many optimization applications and wassuccessfully applied to some practical problems, such asunconstrained numerical optimization [12-15], constrainednumerical optimization [16-17], digital filter design [18],aircraft attitude control [19], and made a series of goodexperimental results. In this paper ABC algorithm has beenemployed to IEEE 30-bus and IEEE-57 bus test systemshaving linear/nonlinear operating constraints, smooth / nonsmooth cost curves under different objective functions. Theobjective functions used in this study are minimization of fuelcost, valve point effect and multi-fuel of generation units. Thepotential and effectiveness of the proposed algorithm aredemonstrated and the results are compared with the existingalgorithms in the literature survey.

2. PROBLEM FORMULATIONThe objective of OPF is to minimize the production cost whilesatisfying all the equality and inequality constraints, and canbe written in the following formMinimizeSubject to:

International Journal of Computer Applications (0975 8887)

u T PG 2 ....PGNG , V G 1....V GNG ,QSh 1 ......,QSh NC ,T1....T NT

In order to demonstrate the effectiveness and robustness of the

proposed algorithm, several cases with different objectives areindicated below.

where

NB

Number of load buses;

NG

Number of generators;

NTL

Number of Transmission Lines;

NT

Number of regulating transformers;

Number of shunt Volt Amperes Reactive (VAR)

compensators.

NC

Minimization of fuel cost: The aim of this type of

problem is to minimize the total fuel cost of all generatingunit which is represented as a quadratic function of its poweroutput and it is formulated as follows:NG

The equality constraint set typically consists of the load flow

equations, which are given below:N

PGi PLi V i V j G ij cos ij B ij sin ij

j 1

NQ Q V V G sin B cos

GiLiijijijijij

j 1

(5)

whereVoltage of i th and j

th

PLi Q Li

Active and Reactive power of i th load bus;

G ij , B ij ,

Conductance, Admittance and Phase difference of

ij

voltages between i th and j

Number of buses.

th

bus.

ai ,bi ,c i : fuel cost

NG

k Q QGi QGilim

NL

kV

VNB

i 1

lim k S S NTL S NTLi 1

NLB

limV NLB

(8)

Non-smooth cost function with Valve-point

effects:

The valve-point boiler of generating units taken in

consideration by adding a sine component to the quadraticcost function. Typically, the fuel cost function of thegenerating units with valve-point is represented as follows [8]:

F ai PGi2 bi PGi c i d i sin e i PGimin PGi

Generator voltage magnitudes, active and reactive power of

i th bus lies between their upper and lower limits as givenbelow:

Minimum and maximum generator voltage

of i th generating unit;

QGimin , QGimax :

Minimum and maximum reactive power of

i th generating unit.

PGimin , PGimax :

Minimum and maximum active power of

i th generating unit.

(9)

d i and e i are the cost coefficients of the unit with valve-point

effects.

(6)

V Gimin , V Gimax :

Piecewise quadratic fuel cost functions

In power system operation conditions, many thermal

generating units may be supplied with multiple fuel sourceslike coal, natural gas and oil. The fuel cost functions of theseunits may be dissevered as piecewise quadratic fuel costfunctions for different fuel types [20]. Thus, the fuel costfunction should be practically expressed as: ai 1PGi2 bi 1PGi C i 1

2 a P b P CFi PGi i 2 Gi i 2 Gi i 2

a P 2 b P C ik Gi ik Gi ik

fuel 1,

PGimin PGi PGi 1

fuel 2,

PGi 1 PGi PGi 2

(10)

fuel k , PGik 1 PGi PGimax

Voltage magnitudes at each bus in the network

V Nmin V N V NmaxThe transmission LinesmaxS NTL S NTL

(7)

In the most of the nonlinear optimization problems, the

constraints are considered by generalizing the objectivefunction using penalty terms [19]. In OPF problem the hardinequalities of P ,V , Q , and, S are added to the objectivefunction and any unfeasible solution obtained is rejected. Theabove penalty function is expressed mathematically asfollows: [20].

Generator constraints:

V Gimin V Gi V Gimax minmax PGi PGi PGi minmaxQGi QGi QGi

coefficients of generator i ; PGi : power generated in (p.u).

i 1

2.2. Inequality Constraints

i 1

Active and Reactive power of i th generator;

f x ,u ai PGi2 bi PGi c i

F f x ,u k P PG 1 PGlim1

bus respectively;

PGi , QGi

Min

Where f : is the total fuel cost ($/hr);

2.1. Equality Constraints

V i ,V j

2.3. Objective Function

The discrete transformer tap settings

T NTmin T NT T NTmax

thwhere aik , bik , and c ik are cost coefficients of the igenerator using the fuel type.[21-22].

3. ARTIFICIAL COLONY BEE

ALGORITHM (ABC)Artificial bee colony (ABC) algorithm is among of newestsimulated evolutionary algorithms. The algorithm was firstlyproposed by Turkish scholar KARABOGA [23]. Three typesof bees are considered in the ABC: employed, onlooker and

46

International Journal of Computer Applications (0975 8887)

Volume 112 No. 1, February 2015scout bees. The number of employed bees is equal to thenumber of food sources and an employed bee is assigned toone of the sources (SN).[24] The position of a food sourcerepresents a possible solution to the optimization problem andthe nectar amount of a food source corresponds to the quality(fitness) of the associated solution [24][35].In the ABC algorithm, each cycle of the search consists ofthree steps: sending the employed bees onto the food sourcesand then measuring their nectar amounts; selecting of the foodsources by the onlookers after sharing the information ofemployed bees and determining the nectar amount of thefoods; determining the scout bees and then sending them ontopossible food sources.[34] At the initialization stage, a set offood source positions are randomly selected by the bees usingthis equationjU i j U min rand

jj U min0,1 . U max

(11)

where j 1,2........, D (D is the number of parameters to be

optimized). The bees in second step search for a new locationin the current position vector neighborhood; search formula isV i j U i j ij U i j U kj(12)

Where k 1,........, N

and j 1,2........, D are randomly

chosen indexes, and k is determined randomly, it has to be

different from i, ij is a random number between [-1, 1].From (13), we can see that as the difference between theparameters of U i j and U kj decreases, the perturbation on theposition U i j decreases, too. Thus, as the search approaches tothe optimum solution in the search space, the step length isadaptively reduced. An onlooker bee chooses a food sourcedepending on the probability value associated with that foodsource, pi calculated by the following expression (13):

Pi

k 1

IEEE 30- bus test System

The standard IEEE 30-bus test system was used to testeffectiveness of ABC algorithm. The test system consists ofsix generating units interconnected with 41 branches of atransmission network to serve a total load of 283.4 MW and126.2 Mvar. The bus data and the branch data are presented inthe reference [25]. Three different types of generator costcurves which are: a quadratic model, a piecewise quadraticmodel and a quadratic model with sine component have beenconsidered as follows:Data : Read system data, unit data, bus-data , line-dataand set the control Parameters of the ABC algorithm

NP :The number of colony size (Number of Foods)

MCN :Maximum Cycle NumberLimit: Maximum number of trial for abandoning a sourcebeginInitializationsfor

k =1

(13)

to NP do

u(k)

random solution byf(u(k)); trial

fk

0;

Cycle =1;While Cycle < MCN do

//

Employed Bees phase :

k =1

u'

to NP do

a new solution produced by Eq

12

f(u')

evaluate new solution using

where Fitni is the fitness value of the solution

which isproportional to the nectar amount of the ith food source. Forminimization problem, Fitni can be calculated using thefollowing expression: 1

Fitni 1 Fi1 Fi

equation 11

end

for

Fitn iSN

Fitn

4. NUMERICAL RESULTS AND

where Fi is the value of the objective function.

In a cycle, after all employed bees and onlooker beescomplete their searches, the algorithm checks to see if there isany exhausted source to be abandoned. Providing that aposition cannot be improved further through limit, then thatfood source is assumed to be abandoned. The food sourceabandoned by its bee is replaced with a new food source U i jrandomly discovered by the scout using the equation (11).Finally memorize the best food source position (solution)achieved, else modify parameters variables by changing theposition of individuals and evaluate fitness (equation (12)) tillmaximum Cycle Number (MCN). The flowchart of ABCalgorithm is drawn in figure 1.

u';

f(u'); trial(k)

fk

0;

elsetrial(k)

trial(k)+1;

end

end//Calculate probabilities for onlooker beesby

equation (13)

// OnlookerK

bees phase

0 ;

0 ;

While t < NP dor

rand(0,1)

// probabilistic

selection

P(k)

47

International Journal of Computer Applications (0975 8887)

Volume 112 No. 1, February 2015if r < P(k) thent

t+1;

u'

a new solution produced by 13

f(u')

evaluate new solution;

if f(u') < fk then

u(k)

u';

f(u'); trial(k)

fk

0;

elsetrial(k)

trial(k)+1;

endendendk

k+1;

if k

NP+1; k

1;

end

// Scout

bees phase

ind={ k : trial(k)=max (trial)

PG11 (MW)

11.9622

12.6419

10.

18.4633

PG13 (MW)

12

12.00

12

19.4613

VG1 (pu)

1.1

1.0421

1.0134

1.1

VG2 (pu)

1.0839

1.0276

0.9849

1.0802

VG5 (pu)

1.0548

1.0169

0.9865

1.0531

VG8 (pu)

1.0573

1.0020

0.97

1.0615

VG11 (pu)

1.1

1.0630

1.0254

1.1

VG13 (pu)

1.1

1.0451

1.1

1.1

T6-9

1.04

0.97

1.01

1.02

T6-10

0.90

0.93

1.1

0.90

T4-12

1.04

0.99

0.90

1.03

T27-28

0.97

0.94

0.90

0.97

Fuel Cost($/h)

799.669

803.9613

930.1114

646.566

Loss(MW)

8.8097

9.8693

14.1364

6.6891

|Vi-Vref|

1.271

0.0193

0.5815

1.1172

if trial(ind )>Limit then

Table 2. Comparison of the simulation results for CASE-1

u(ind)

random solution by Eq

12Methods

find = f(u(ind))trial(ind )

0;

end

Best

Average

Worst

ABC

799. 669

799.766

800.063

EADDE [27]

800.2041

800.2412

800.2748

MDE[28]

802,376

802,382

802,404

MCN = MCN+1;

Fig. 1. Flowchart for the ABC-Algorithm

1. Case.1: Quadratic cost curve model

To demonstrate the consistency and robustness of theproposed algorithm, 30 independent runs for each case wereconducted performed for reaching the optimal. In this case theunit cost curves are represented by quadratic functions (1).The voltage magnitude of generator (PV) is set between 0.951.1. The maximum and minimum voltages of all load buses(PQ) are considered to be 1.05 - 0.95 in pu. The operatingrange of all transformers is set between 0.90 -1.1 with anadjustable step size of 0.01p.u.The solution details for the minimum cost are provided in TableI, the average cost of solution obtained was 799.766$/hr with theminimum being 799.66 $/hr 8.8097 MW losses and maximumof 800.063$/hr. Fig 2 shows the convergence curve of ABCOPF for the trial run that produced the minimum cost solution. Itis important to note that all control and state variables remainedwithin their permissible limits.

BBO[29]

799,1116

799,1985

799,2042

LDI-PSO[20]

800.7398

801.5576

803.8698

GSA[20]

798.675143a

798.913128

799.028419

825819816813810807804801798795

Table 1. Best control variables settings for different test

case

case 1

822

Fuel Cost ($/hr)

end

30

60

90

120

150

Iteration

Fig. 2. Convergence curve of the OPF-ABC to Case 1

Variable

Case 1

Case 2

Case 3

Case 4

PG1 (MW)

177.3762

175.6484

199.5897

139.9926

PG2 (MW)

48.5834

48.8422

50.9467

54.9704

PG5 (MW)

21.3299

21.6699

15

23.9236

PG8 (MW)

20.958

22.4669

10

33.2779

2. Case.2: voltage profile improvement

Bus voltage is one of the most important security and servicequality indices. Considering only cost-based objectives inOPF problem may result in a feasible solution that hasunattractive voltage profile. So, in this case a two-foldobjective function will be considered in order to minimize thefuel cost and improve voltage profile by minimizing the load

48

International Journal of Computer Applications (0975 8887)

Volume 112 No. 1, February 2015bus voltage deviations from 1.0 per unit. The objectivefunction can be expressed [19]:f x ,u

NPQ

NG

ai PGi2

bi PGi c i

i 1

1.0

(18)

i 1

where is a suitable weighting factor, to be selected by the

user. Value of in two test systems is chosen as 100. Theoptimal setting of the control variables are given in Table I.Voltage profile in this case is compared to that of case (1) asshown in Fig 3; It is quite evident that the voltage profile isimproved compared to that of Case (1), and if somebodythrow a glance at Fig 3 remark clearly that the voltagemagnitudes in load buses: 3,4,6,7,12,14, 28 and 29 relatedat case (1), overtaken the upper limit fixed at 1.05 in pu,with 2.29%, 1.67%, 0.81%, 0.65%, 1.33%, 0.25%, and 0.33%,respectively, this is justified by the strategy of penaltyfunction which presents no problems when enforcing softlimits.However in case (2), all overtaking signaledpreviously in case (1) are closer at 1 pu. (See Fig 3).It is decreased from 1.271 pu in Case (1) to 0.0193 pu in case(2). The result obtained from the proposed algorithm reduces98.4815% in this case. Table III summarizes the comparisonresults of the voltage profile improvement. Table IV list liststhe statistical results in terms of the best, mean, and worstvoltage deviation. From these results, it is clear that ABCobtained a lower value and has a better than those reported inthe literature.

3. Case 3: Quadratic cost curve model with sine

ComponentIn this case, the generating units of buses 1 and 2 areconsidered to have the valve-point effects on theircharacteristics. The cost coefficients for these units are givenin Reference [27]. The fuel cost coefficients of the restgenerators have the same values as a case (1). The voltagemagnitude of generator is set to 0.95 V i 1.1 .Themaximum and minimum voltages of all load buses areconsidered to be 1.05 - 0.95 in pu. Limits of transformer tapsettings are taken as 0.90 V i 1.1 p.u with an adjustablestep size of 0.01p.u. The set of optimal solutions of controlvariables are presented in Table I. The comparison results arepresented in Table V. From simulation results it is veryobvious that ABC algorithm has better quality of solutionsthan EP, IEP, MDE and BBO. It is clear that the minimumfuel cost obtained from the proposed algorithm is 929.902 $/h

49

International Journal of Computer Applications (0975 8887)

Volume 112 No. 1, February 2015with an average cost of 930.971 $/h and a maximum cost of932.428 $/h, which is less than MDE algorithm and is morethan BBO algorithm. But the sum of real power of generatingunits was given as 294.464MW in BBO approach and realpower loss was 12.18MW whereas load was 283.4 MW. Sopower generation is not matching load plus losses. Thisapproach did not meet the load demand for this case [20]. Theconvergence curve of ABC algorithm for the OPF problemwith minimum fuel cost is shown Fig 4.The results obtainedconfirm the ability of the proposed ABC algorithm to findaccurate OPF solutions in this case study.Table 5. Comparison of the simulation results for CASE-3Voltage profile improvement

Table 6. Comparison of the simulation results for CASE-4

CASE-4 PiecewiseOutputs

DGA[8]

DE[28]

IEP [30]

MDE[28]

ABC

PG1

139.95

139.96

139,996

140.00

140.00

PG2

55.00

54.984

54.9849

55.00

55.00

PG5

23.28

23.910

23.2558

24.000

25.9317

PG8

34.36

34.291

34.2794

34.989

34.3422

MethodsBest

Average

Worst

PG11

19.16

21.161

17.5906

18.044

16.6520

ABC

929.9021

930.971

932.428

PG13

18.85

16.202

20.7012

18.462

18.1906

BBO[30]

919.7647

919.8389

919.8876

290.509

290.808

290,495

290.116

MDE[32]

930.793

942.501

954.073648.38

649.312

647.846

646.890

7.4081

7.095

7.0527

Total290.60(MW)Fuel Cost

EP [31]

955.508

IEP [31]

957.709

953.573

956.460

959.379Losses(MW)

958.263

Fuel Cost ($/hr)

7.204

7.109

IEEE 57- bus test System

1100Case 3

10801060104010201000980960940920900

648.40

30

60

90

120

150

Iteration

Fig. 4. Convergence curve of the OPF-ABC to Case 3

The cost coefficients for these units are given in Ref [9].The cost characteristics of the first and second generators aredefined in equation (10). The proposed algorithm is applied tothis case considering the limit of controls variables has thesame limits as a third Case. The results obtained optimalsettings of control variables for this case study are listed inTable I, which shows that the ABC has best solution forminimizing of fuel cost in the OPF problem. The best fuelcost result obtained from the ABC approach is compared withother algorithms in Table VI. The average cost of solutionobtained was 648.6970$/hr with the minimum being646.891$/hr and maximum of 650.9820$/hr. According toresults of the third and fourth cases, it appear that ABCalgorithm has better results compared to other algorithmspreviously reported in the literature.

In order to verify the robustness and efficiency of the

proposed algorithm to the larger power system, the algorithmwas tested and examined to standard IEEE 57-bus test system.The system has totally 27 variables to be optimized, including7 generators, 17 transformers (treated as tap changer), and 3capacitor banks installed at buses 18, 25 and 53 respectively.The total load demand of system is 1250.8 MW and 336.4Mvar under the base of 100 MVA. The bus 1 is selected asslack bus. The single line diagram of this system and the busdata and line data can be retrieved at MATPOWER [33]. Themaximum and minimum voltages of all buses are consideredto be 0.95 1.1 in p.u. The operating range of all transformersis set between 0.90 -1.1. The minimum and the maximum ofshunt capacitor banks are 0.0 and 0.3 in p.u. The controlparameter settings of the ABC algorithm related to this casestudy are provided in Table VII below:Table 7. Control parameter settingsParameter

IEEE 57-Bus

Population size (NP)

40

Max. Cycle number (MCN)

200

Penalty factor of slack bus real power (KP)

1000

Penalty factor of reactive power (KQ)

100

Penalty factor of voltage magnitudes (KV)

100.000

Penalty factor of transmission line loadings (KS)

50

50

International Journal of Computer Applications (0975 8887)

Volume 112 No. 1, February 20155

The set of optimal solutions of control variables from

the proposed algorithm are presented in Table VIII. Fromthis Table, it is clear that the best solution of presentedresult is that of the GSA marked "a", and he is much lessthan solution obtained by ABC algorithm but is indeed aninfeasible solution, since there exist bus voltage magnitudeviolations at buses 18,19, 20, 26, 27, 28, 29, 30, 31, 32,33,42,51,56 and 57 and the true value for the total fuel costcorresponding to the set of optimal solutions of controlvariables reported by GSA is 45621.4035 $/hr.

x 10

10

Case 2, IEEE 57

98765432

The obtained results are compared with that of the particle

swarm optimization (PSO), Cuckoo Optimization Algorithm(COA), LDI-PSO, EADDE, GSA and MATPOWER. Thiscomparison confirms the aptitude of the artificial bee colonyalgorithm to locate de global solution. Fig 5 shows theconvergence curve related to improvement of voltage profile(case 2). Also, it is important to note that all optimizationvariables remained within their permissible limits without anyviolations.

Table 10. Maximum power flow limit of transmission line

Total Fuel Cost ($/hr

41705.3

41827.4

P loss (MW)

15.3215

16.8817

Voltage Deviation (VD)

1.9220

0.3013

100

5. CONCLUSIONA simple Artificial Bee Colony algorithm is proposed to solvethe OPF problem under different formulations andconsidering different objectives function. The performance ofthe proposed ABC was tested on the IEEE 30-bus test andIEEE 57 test systems. The results obtained using the ABCalgorithm were compared to other methods previouslyreported in the literature.

51

International Journal of Computer Applications (0975 8887)

Volume 112 No. 1, February 2015The comparison verifies the influentially of the proposedABC approach over stochastic techniques in terms of solutionquality for the OPF problem and confirmed its potential forsolving a most nonlinear problems.